Running Wires From Basement Question

I've got experience running cables from basement to the first and second floors from my last house, but my new home has me stuck. My last home, I had engineered rim joists which means that to get a cable into the wall above, all I had to do was drill down into the sill plate on the first floor and down through the top of the 2x4 that formed the top of the engineered rim joist...pretty simple. my new house has a different construction that makes this more difficult. The picture below shows the general cross section of what I am dealing with. Basically, there is a sill plate on top of the concrete basement wall followed by 2 doubled up 2x10 rim joists followed by another sill plate. The first level sub floor then sites on top of that and the framed first level wall on top of that. The basement joists are joined to the rim joists with joist hangers. Drilling straight down from the top in the framed wall cavity will not work since I will drill through the sill plate and into the doubled 2x10s...the hole will not exit into the basement which would happen if that was an engineered beam. How do I drill? Is it OK to drill on an angle from the top such that I pop the hole out on the side of the rim joists? Thanks for the help.

At least one company makes 4' drill augers that have a flexible shaft designed for running wires. The drill bit has a hole drilled through it so after you drill through, you can use it as a fish to pull back the wire. I've seen them at HD. Not sure how well they work, but you'd definately have the length and flexibility you'd need. I think they also sell extensions for them, so you could go quite a ways with the thing. Tricky to get started, maybe, but it has a screw on the end to bite and pull the auger through the wood.